Canonical Forms for Reachable Systems over Von Neumann Regular Rings

Abstract

If (A,B) is a reachable linear system over a commutative von Neumann regular ring R, a finite collection of idempotent elements is defined, constituting a complete set of invariants for the feedback equivalence. This collection allows us to construct explicitly a canonical form. Relations are given among this set of idempotents and various other families of feedback invariants. For systems of fixed sizes, the set of feedback equivalent classes of reachable systems is put into 1-1 correspondence with an appropriate partition of Spec(R) into open and closed sets. Furthermore, it is proved that a commutative ring R is von Neumann regular if and only if every reachable system over R is a finite direct sum of Brunovsky systems, for a suitable decomposition of R.